A note about invariant SKT structures and generalized Kähler structures on flag manifolds

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1 arxiv: v4 [math.dg] 25 Feb 2012 A note about invariant SKT structures and generalized Kähler structures on flag manifolds Dmitri V. Alekseevsky and Liana David October 29, 2018 Abstract: We prove that any invariant strong Kähler structure with torsion (SKT structure) on a flag manifold M = G/K of a semisimple compact Lie group G is Kähler. As an application we describe invariant generalized Kähler structures on M Mathematics Subject Classification: 53D18. 1 Introduction A Hermitian manifold (M,g,J) admits a unique connection B (called the Bismut connection) which preserves the metric g and the complex structure J and has a skew-symmetric torsion tensor c := g(,t B (, )), where T B is the torsion of B. The 3-form c can be expressed in terms of the Kähler form ω = g J by c = Jdω := dω(j,j,j ). The manifold (M,g,J) is called strong Kähler with torsion (SKT) if the torsion 3-form c is closed, or, equivalently, ω = 0. SKT manifolds are a natural generalization of Kähler manifolds and many results from Kähler geometry can be generalized to SKT geometry, see e.g. [3, 4, 6, 7]. SKT geometry is also closely related to generalized Kähler geometry, which was recently introduced by N. Hitchin [12] and appeared before in physics as the geometry of the target space of N = (2,2) supersymmetric nonlinear sigma models, see e.g. [8, 13]. 1

2 A generalized Kähler structure on a manifold M is a pair (J 1,J 2 ) of commuting generalized complex structures such that the symmetric bilinear form (J 1 J 2, ) is positive definite, where (, ) is the standard scalar product of neutral signature of the generalized tangent bundle TM = TM T M. (For the definition and basic facts about generalized complex structures, see e.g. [9]). It was shown by M. Gualtieri [9, 10] that a generalized Kähler structure on a manifold can be described in classical terms as a bi-hermitian structure (g,j +,J,b) in the sense of [8], i.e. a pair (g,j + ), (g,j ) of SKT structures with common metric g and a 2-form b (called in the physical literature the b-field) such that db = J + dω + = J dω, (1) where ω ± = g J ± are Kähler forms. Let G be a semisimple compact Lie group and M = G/K a flag manifold, i.e. an adjoint orbit of G. In this note we describe invariant SKT structures and invariant generalized Kähler structures on M, as follows. Theorem 1. Any invariant SKT structure (g,j) on a flag manifold M = G/K is Kähler, i.e. the Kähler form ω = g J is closed. The description of invariant Kähler structures on flag manifolds is well known, see e.g. [1, 2] and Section 2 below. Corollary 2. Let (g,j +,J,b) be an invariant bi-hermitian structure in the sense of [8] on a flag manifold M = G/K (which defines a generalized Kähler structure (J 1,J 2 ) via Gualtieri s correspondence). Then g is an invariant Kähler metric, J +,J are two parallel invariant complex structures and b is any closed invariant 2-form. If the group G is simple, then J + = J or J + = J. The note is organized as follows. In Section 2 we fix our conventions and we recall the basic facts on the geometry of flag manifolds and, in particular, the description of invariant Hermitian and Kähler structures [1, 2]. With these preliminaries, Theorem 1 and Corollary 2 will be proved in Section 3. Acknowledgements. D.V.A thanks the University of Hamburg for hospitality and financial support. L.D. acknowledges financial support from a grant of the Romanian National Authority for Scientific Research, CNCS- UEFISCDI, project number PN-II-ID-PCE

3 2 Preliminary material Basic facts about flag manifolds. A flag manifold of a semisimple compact Lie group G is an adjoint orbit M = Ad G (h 0 ) G/K of an element h 0 of the Lie algebra of G. We denote by g, k the complex Lie algebras associated with the groups G, K respectively, and we fix a Cartan subalgebra h of k. We denote by R,R 0 the root systems of g, k with respect to h and we set R := R\R 0. We write the root space decomposition of g as g = k+m = (h+ α R 0 g α )+ α R g α and we identify the vector space m = α R g α with the complexification of the tangent space T h0 M. Let E α g α be root vectors of a Weyl basis. Thus, E α,e α = 1, α R (where X,Y := tr(ad X ad Y ) denotes the Killing form of g) and N α, β = N αβ, α,β R (2) where N αβ are the structure constants defined by [E α,e β ] = N αβ E α+β, α,β R. (3) The Lie algebra of G is the fixed point set g τ of the compact antiinvolution τ, whichpreserves thecartansubalgebrahandsendse α to E α, for any α R. It is given by g τ = ih R + α Rspan{E α E α,i(e α +E α )} where h R = span{h α := [E α,e α ],α R} is a real form of h. Note that β(h α ) = β([e α,e α ]) = β,α where, denotes also the scalar product on h induced by the Killing form. 3

4 2.1 Invariant complex structures on M = G/K We fix a system Π 0 of simple roots of R 0 and we extend it to a system Π = Π 0 Π of simple roots of R. We denote by R 0,R + + = R 0 + R + the corresponding systems of positive roots. A decomposition m = m + +m = g α + g α α R + α R + defines an Ad K -invariant complex structure J on T h0 M = m τ, such that J m ± = ±iid. We extend it to an invariant complex structure on M, also denoted by J. We will refer to R + and Π as the set of positive roots, respectively the set of simple roots, of J. It is known that any invariant complex structure on M can be obtained by this construction [1, 15]. 2.2 T-roots and isotropy decomposition Let z = it h be the center of the stability subalgebra k τ. The restriction of the roots from R h to the subspace t are called T-roots. Denote by κ : R R T, α α t the natural projection onto the set R T of T-roots. Note that α t = 0 for any α R 0. Any T-root ξ defines an Ad K -invariant subspace m ξ := α R,κ(α)=ξ of the complexified tangent space m and m = ξ R T m ξ isadirectsumdecompositionintonon-equivalentirreduciblead K -submodules. g α 4

5 2.3 Invariant metrics and Hermitian structures We denote by ω α g the 1-forms dual to E α, α R, i.e. ω α (E β ) = δ αβ, ω α h = 0. (4) Any invariant Riemannian metric on M is defined by an Ad K -invariant Euclidean metric g on m τ, whose complex linear extension has the form g = 1 g α ω α ω α (5) 2 α R where ω α ω α = ω α ω α + ω α ω α is the symmetric product and g ξ, ξ R T, is a system of positive constants associated to the T-roots, g ξ = g ξ for any ξ R T and g α := g κ(α). Note that the restriction of g to m ξ is proportional to the restriction of the Killing form, with coefficient of proportionality g ξ. Any such metric g is Hermitian with respect to any invariant complex structure J and the corresponding Kähler form is given by ω = i g α ω α ω α (6) α R + where R + is the set of positive roots of J and in our conventions ω α ω α = ω α ω α ω α ω α. 2.4 Invariant Kähler structures Any invariant symplectic form ω on M compatible with an invariant complex structure J as above (i.e. such that g := ω J is positive definite) is associated to a 1-form σ t such that σ,α i > 0 for any α i Π (the set of simple roots of J). As a form on m, it is given by ω = ω σ := i σ,α ω α ω α. α R + The associated Kähler metric g has the coefficients g α = g κ(α) = σ,α, which, obviously, satisfy the following linearity property: g α+β = g α +g β, α,β,α+β R + (7) 5

6 In particular, if Π = {α 1,,α m } and then To summarize, we get: R α k 1 α 1 + +k m α m (modr 0 ) g α = k 1 g α1 + +k m g αm. Proposition 3. [1] An invariant Hermitian structure (g, J) on M is Kähler if and only if the coefficients g α associated to g by (5) satisfy the linearity property: if α,β,α + β R +, then g α+β = g α + g β. Here R + is the set of positive roots of J. 2.5 The formula for the exterior derivative An invariant k-form on M = G/K can be considered as an Ad K -invariant k-form ω on the Lie algebra g such that i k (ω) = 0. We recall the standard Koszul formula for the exterior differential dω: (dω)(x 0,,X k ) = i<j ( 1) i+j ω([x i,x j ],X 1,, X i,, X j,,x k ), for any X i m g. In (8) the hat means that the term is omitted. (8) 3 Proof of our main results We now prove Theorem 1 and Corollary 2. We preserve the notations from the previous sections. Let (g, J) be an invariant Hermitian structure on a flag manifold M = G/K. Let g α = g k(α) the positive numbers associated to g and R +, Π the set of positive (respectively, simple) roots of J, like before. Let ω = g J be the Kähler form. To prove the theorem, we have to check that if the form Jdω is closed, then g α satisfy the linearity property (7). We define the sign ǫ α of a root α R = R + ( R +) by ǫ α = ±1 if α ±R +. Note that ǫ α depends only on κ(α). Now we calculate dω and Jdω on basic vectors, as follows: 6

7 Lemma 4. i) dω(e α,e β,e γ ) = 0 if α+β +γ 0 (9) and dω(e α,e β,e (α+β) ) = in αβ (ǫ α g α +ǫ β g β ǫ α+β g α+β ). (10) ii) (Jdω)(E α,e β,e (α+β) ) = N αβ (ǫ β ǫ α+β g α +ǫ α ǫ α+β g β ǫ α ǫ β g α+β ). (11) Proof. Relation (9) follows from (6) and (8). Relation (10) follows from (6), (8) and the following property of N αβ (see Chapter 5 of [11]): if α,β,γ R are such that α+β +γ = 0, then N αβ = N βγ = N γα. (12) Relation (11) follows from (10) and JE α = iǫ α E α for any α R. Lemma 5. Suppose that (g,j) is a SKT structure, i.e. d(jdω) = 0. Then N 2 αβ(g α+β g α g β )+ǫ α β N 2 α, β(ǫ α β g α β g α +g β ) = 0 (13) for any α,β R +, where we assume that ǫ α β = 0 if α β / R. Proof. By a direct computation, we find 1 2 d(jdω)(e α,e β,e α,e β ) = N 2 αβ (g α+β g α g β ) This relation implies our claim. For any root +ǫ α β N 2 α, β(ǫ α β g α β g α +g β ). R + α k 1α 1 + +k m α m (modr + 0 ), α i Π, we define the length of α as l(α) = m i=1 k i. Note that l(α) depends only on the projection κ(α) of α onto t. 7

8 Proof of Theorem 1. By Proposition 3 we have to check that g α+β = g α +g β, (14) for any α,β R + such that α+β R +. We use induction on the length of γ = α+β R +. Suppose first that γ = α +β R + has length two. Then α β / R, hence ǫ α β = 0. Identity (13) implies (14). Suppose now that (14) holds for all γ = α + β R + with l(γ) k. Let γ R + with l(γ) = k+1 and suppose that γ = α+β, where α,β R +. We have to show that g γ = g α +g β. (15) If α β / R, our previous argument shows that (15) holds. Suppose now that α β R. Without loss of generality, we may assume that α β R +. Then α = (α β) + β is a decomposition of the root α into a sum of two roots from R +. Since α has length k, our inductive assumption implies that g α = g α β +g β. Thus the second term of the identity (13) vanishes and we obtain (15). This concludes the proof of Theorem 1. Proof of Corollary 2. Let (g,j +,J,b) be a G-invariant bi-hermitian structure in the sense of [8] onaflag manifold M = G/K. Then, by Theorem 1, (g,j ± ) are two Kähler structures and hence the b-field b is closed. The complexstructuresj ± areparallelwithrespect tothelevi-civitaconnection. If the group G is simple, the Kähler metric g is irreducible. The endomorphism A = J 1 J 2 is symmetric with respect to g and parallel. An easy argument which uses the irreducibility of g shows that J 1 = J 2 or J 1 = J 2. This concludes the proof of Corollary 2. References [1] D. V. Alekseevsky, A. M. Perelomov: Invariant Kähler-Einstein metrics on compact homogeneous spaces, Funct. Anal. Applic. 20 (1986), no. 3, p [2] D. V. Alekseevsky: Flag manifolds, Preprint ESI, (1997) no. 415, 32 p.; Zbornik Radova, book 6 (14),(1997), 11 Yugoslav. Seminar, Beograd, p [3] N. Enrietti, A. Fino, L. Vezzoni: Tamed Symplectic forms and SKT metrics, math.dg/ ; to appear in J. Symplectic Geometry. 8

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10 LIANA DAVID: Institute of Mathematics Simion Stoilow of the Romanian Academy; Calea Grivitei nr. 21, Sector 1, Bucharest, Romania; 10